Simplicity in simplicial phase space

TL;DR

This work develops a discrete phase-space formulation of Plebanski gravity on a simplicial manifold, detailing both gauge-variant and gauge-invariant implementations of the primary and secondary simplicity constraints. It demonstrates that, under non-degeneracy, these constraints imply the diagonal, cross, and edge simplicity conditions and give rise to gluing constraints that fix the spatial Levi-Civita connection and ensure consistent 4d geometry, reducing the phase space to a compact set of geometric degrees of freedom described by areas and 4d-dihedral angles. A concrete boundary-of-a-4-simplex example illustrates constraint counting and the reduction to a 20-dimensional reduced phase space parameterised by 10 areas and 10 4d-dihedral angles. The analysis also clarifies the role of degenerate configurations and their impact on spin-foam models, arguing for the necessity of both primary and secondary constraints to obtain the correct boundary Hilbert space and dynamics, with implications for connecting spin foams to canonical LQG and for future canonical quantisation work.

Abstract

A key point in the spin foam approach to quantum gravity is the implementation of simplicity constraints in the partition functions of the models. Here, we discuss the imposition of these constraints in a phase space setting corresponding to simplicial geometries. On the one hand, this could serve as a starting point for a derivation of spin foam models by canonical quantisation. On the other, it elucidates the interpretation of the boundary Hilbert space that arises in spin foam models. More precisely, we discuss different versions of the simplicity constraints, namely gauge-variant and gauge-invariant versions. In the gauge-variant version, the primary and secondary simplicity constraints take a similar form to the reality conditions known already in the context of (complex) Ashtekar variables. Subsequently, we describe the effect of these primary and secondary simplicity constraints on gauge-invariant variables. This allows us to illustrate their equivalence to the so-called diagonal, cross and edge simplicity constraints, which are the gauge-invariant versions of the simplicity constraints. In particular, we clarify how the so-called gluing conditions arise from the secondary simplicity constraints. Finally, we discuss the significance of degenerate configurations, and the ramifications of our work in a broader setting.

Figures (1)

• Figure 1: Two tetrahedra sharing a triangle and the associated 2d dihedral angles and edge vectors.